I know that it is generally a bad idea to compute the inverse matrix directly. However, if it is necessary to compute the inverse of an ill-conditioned invertible dense matrix, then what can I try?
For example, I know that scaling a matrix may decrease the condition number of it. This may be an example of preconditiong, but I don't know much about the theory of preconditioners.
But I can't find the general principle to compute the inverse because I am new to this field. What properties of the matrix should I check to reduce the condition number? Is there any strategy, or, any books/papers which explain more accurate methods to compute the inverse directly?
(I don't know if there is a short answer of this. Feel free to close this question as too broad.)